Don't throw that sextant away!

[HenryC]

A circle has 360 degrees because thousands of years ago the Babylonians thought the year was 360 days long.  This made sense to them, but we’ve been stuck with this number “360” as if it was really fundamental.  Actually, it is a historical accident.  To make matters worse, we have divided each degree into 60 minutes and each minute into 60 seconds, making calculations with angles about as mystical and awkward as trying to make change in England before their currency went metric. 

The smart and natural way to describe angles is the way mathematicians and physicists do it, with radian measure.  This makes calculus and angular momentum calculations very easy.  So just what is a radian?  There are 2 times pi radians in a full 360 degree circle, pi radians in a half-circle (180 degrees) and pi/2 radians in a 90 degree right angle.  (Forget about the explanations,  just remember a radian is equal to about 57.3 degrees--for eyeball calculations, use 60 degrees, it’s close enough and easier to remember.)

So why bother, other than to keep mathematicians and physicists happy?  Because the radian is the angle of arc of a circle equal to a length of arc of one radius on that circle--if you cut off one radian of arc from a circle, and straighten it out, it will be equal in length to one radius of that circle.  Phrasing it yet another way, if you have a circle of radius x feet, a radian angle of that circle will also be x feet long.  This allows you to relate angles to distances easily, which is an exercise sailors are always doing! 

A quick example:  If a distant smokestack appears to be 1/159 of a radian tall, then its distance from you is 159 times as far away from you in feet as the smokestack is high in feet.  Since most vertical landmarks on a chart have their heights printed, this means you can get a range to them if you can just measure the angle.  And this is particularly easy since we know a radian is equal to approximately 60 degrees (angles are hard to measure accurately so we can afford to be sloppy for the sake of simplicity).

On a boat, we measure angles with a sextant  which is accurate to about a minute (1/60  degree) of arc.  So if we measure a vertical object whose vertical size is charted (say the center span of the bridge is 210 feet over the water) and derive an angle of 32 minutes;  32 minutes is 32/60 of a degree and each degree is roughly 1/60 of a radian.  So the center span is about 32/3600 of a radian in height. Because of the way we have defined the radian, 32/3600 of the distance to the bridge is 210 feet.  The distance of the bridge is then 210 x 3600/32, or 23625 feet away.  That works out to (roughly) 4 nautical miles.

I’ve worked it out long hand and used round-offs so you can follow my logic and convince yourself, and so you can figure it out for yourself if you forget the precise formula below.

Distance (nautical miles) = 0.5658 x Height (feet) / Angle (minutes of arc)

So if we redo our example

Distance = 0.5658 x 210/32 = 3.71 nautical miles

[HenryC]

When using a sextant as a range finder, keep in mind that the technique does not work for nearby objects (less than a half-mile away).  The index and horizon mirrors of a sextant are offset by about six inches, so a parallax error is introduced into the measurement which is large enough to corrupt the angle measured for nearby objects.

Incidentally, since a sextant can accurately measure very small angles, it can be used as a range finder, to determine ship-to-ship distances when steaming in formation with other vessels when the fleet is steaming in stealth (radar-silent) mode.  Another application is to combine a compass bearing with a sextant range to derive a fairly good estimated position off a single distant landmark, or two landmarks too close together to give a good crossing fix.